How to Find the Conjugate of a Complex Number

Here you will learn how to find the conjugate of a complex number and properties of conjugate with examples.

Let’s begin –

Let z = a + ib be a complex number. Then the conjugate of z is denoted by \(\bar{z}\) and is equal to a – ib.

Thus, z = a + ib \(\implies\) \(\bar{z}\) = a – ib

It follows from this definition that the conjugate of a complex number is obtained by replacing i by -i.

For Example : If z = 3 + 4i, then \(\bar{z}\) = 3 – 4i.

If \(z\), \(z_1\), \(z_2\) are complex numbers, then

(i) \(\bar{\bar{z}}\) = z

(ii) z + \(\bar{z}\) = 2 Re (z)

(iii) z – \(\bar{z}\) = 2i Im (z)

(iv) z = \(\bar{z}\) \(\iff\) z is purely real

(v) z + \(\bar{z}\) = 0 \(\implies\) z is purely imaginary

(vi) z\(\bar{z}\) = \([Re (z)]^2\) + \([Im (z)]^2\)

(vii) \(\bar{z_1 + z_2}\) = \(\bar{z_1}\) + \(\bar{z_2}\)

(viii) \(\bar{z_1 – z_2}\) = \(\bar{z_1}\) – \(\bar{z_2}\)

(ix) \(\bar{z_1z_2}\) = \(\bar{z_1}\) \(\bar{z_2}\)

(x) \(\bar{z_1\over z_2}\) = \(\bar{z_1}\over \bar{z_2}\)

Example : Multiply 3 – 2i by its conjugate.

Solution : The conjugate of 3 – 2i is 3 + 2i.

Hence, required product is = (3 – 2i)(3 + 2i) = \(9 – 4i^2\) = 9 + 4 = 13

Example : Find the conjugate of \(1\over 3 + 4i\).

Solution : Let z = \(1\over 3 + 4i\). Then,

z = \(1\over 3 + 4i\) \(\times\) \(3 – 4i\over 3 – 4i\) = \(3 – 4i\over 9 + 16\) = \({3\over 25} – {4\over 25}i\)

\(\therefore\) Conjugate of z is \(\bar{z}\) = \({3\over 25} – {4\over 25}i\).